Performance Bounds on Pliable Index Coding Using Absent Receivers

TL;DR

This paper establishes bounds on the optimal broadcast rate for general pliable index coding instances, introducing a novel decoding chain algorithm and explicitly characterizing rates for specific cases with absent receivers.

Contribution

It introduces a new decoding chain algorithm for pliable index coding and derives explicit bounds for instances with up to four absent receivers or specific side-information patterns.

Findings

Lower bounds depend on absent receivers and their side-information patterns.

Explicit optimal broadcast rates are characterized for small and structured instances.

The algorithm provides a framework for analyzing more general pliable index coding problems.

Abstract

We characterise bounds on the optimal broadcast rate for a few classes of pliable-index-coding instances. Unlike the majority of currently solved instances, which belong to a special class where all receivers with a certain side-information cardinality are either present or absent, we consider more general instances without this constraint. We devise a novel algorithm that constructs a decoding chain by iteratively adding a message that can be decoded by a receiver whose side information is already in the chain. If the decoding chain cannot proceed due to the absence of a receiver with the required messages, we skip a message by adding it to the chain regardless. We prove that a lower bound on the optimal broadcast rate is a function of the number of skipped messages, across all possible decoding choices of the receivers and any realisation of the algorithm for each decoding choice. While this result is not computationally feasible in isolation, it serves as a basis for deriving explicit lower bounds on the broadcast rate for specific classes of pliable-index-coding instances. These lower bounds depend on the number of absent receivers or the pattern of their side-information sets. Specifically, we explicitly characterise the optimal broadcast rate for instances with up to and including four absent receivers with any side-information pattern, as well as instances where the side-information sets are nested in particular ways.